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Frequency-Based Reduced Models from Purely Time-Domain Data via Data Informativity

Michael S. Ackermann, Serkan Gugercin

TL;DR

The data informativity approach to moment matching provides a powerful new framework for recovering the required frequency data from a single time-domain trajectory, resulting in vastly improved conditioning of the associated linear systems, an error indicator, and removal of an assumption that the system order is known.

Abstract

Frequency-based methods have been successfully employed in creating high fidelity data-driven reduced order models (DDROMs) for linear dynamical systems. These methods require access to values (and sometimes derivatives) of the frequency-response function (transfer function) in the complex plane. These frequency domain values can at times be costly or difficult to obtain (especially if the method of choice requires resampling); instead one may have access to only time-domain input-output data. The data informativity approach to moment matching provides a powerful new framework for recovering the required frequency data from a single time-domain trajectory. In this work, we analyze and extend upon this framework, resulting in vastly improved conditioning of the associated linear systems, an error indicator, and removal of an assumption that the system order is known. This analysis leads to a robust algorithm for recovering frequency information from time-domain data, suitable for large scale systems. We demonstrate the effectiveness of our algorithm by forming frequency based DDROMs from time-domain data of several dynamical systems.

Frequency-Based Reduced Models from Purely Time-Domain Data via Data Informativity

TL;DR

The data informativity approach to moment matching provides a powerful new framework for recovering the required frequency data from a single time-domain trajectory, resulting in vastly improved conditioning of the associated linear systems, an error indicator, and removal of an assumption that the system order is known.

Abstract

Frequency-based methods have been successfully employed in creating high fidelity data-driven reduced order models (DDROMs) for linear dynamical systems. These methods require access to values (and sometimes derivatives) of the frequency-response function (transfer function) in the complex plane. These frequency domain values can at times be costly or difficult to obtain (especially if the method of choice requires resampling); instead one may have access to only time-domain input-output data. The data informativity approach to moment matching provides a powerful new framework for recovering the required frequency data from a single time-domain trajectory. In this work, we analyze and extend upon this framework, resulting in vastly improved conditioning of the associated linear systems, an error indicator, and removal of an assumption that the system order is known. This analysis leads to a robust algorithm for recovering frequency information from time-domain data, suitable for large scale systems. We demonstrate the effectiveness of our algorithm by forming frequency based DDROMs from time-domain data of several dynamical systems.
Paper Structure (15 sections, 5 theorems, 59 equations, 8 figures, 3 tables, 1 algorithm)

This paper contains 15 sections, 5 theorems, 59 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The data informativity framework for moment matching
  3. Recovering transfer function values from time-domain data
  4. Theoretical and algorithmic analysis for a robust implementation of data informativity for moment matching
  5. Improving and predicting conditioning
  6. Orthogonalization
  7. Revisiting rank calculations
  8. Windowing and error estimation
  9. Flexibility in choosing system order
  10. Algorithm
  11. Numerical results
  12. Synthetic example
  13. Heat Model
  14. Penzl's example
  15. Conclusions

Key Result

Theorem 2.4

\newlabelthm:ExistUniqueOrig0 Assume access to input data $\mathbb{U}$ and output data $\mathbb{Y}$. Let $n$ be the order of the system and let $\sigma, M_0 \in \mathbb C$. Then the inclusion $\Sigma^n_{(\mathbb{U},\mathbb{Y})} \subseteq \Sigma_{\sigma,M_0}^{n,0}$ holds if and only if there exists Moreover, the data $(\mathbb{U}$,$\mathbb{Y})$ are informative for interpolation at $\sigma$ (i.e.,

Figures (8)

  • Figure 1: Singular values of $\mathbf{G}_{n}\mathbf{z}(\sigma)$, $\mathbf{U}_c\mathbf{z}(\sigma)$, and $\mathbf{U}\mathbf{z}(\sigma)$.
  • Figure 1: Frequency responses ((a) and (c)) and point-wise relative errors (plots (b) and (d)) of DDROMs approximating $H_2$, constructed using frequency data recovered from time-domain data via \ref{['alg:DataInform']} ((a) and (b)) and true frequency data ((c) and (d)).
  • Figure 2: The normalized standard deviation ($s_W$) of $\{M_{0,\ell_i}(e^{\mathbf{i}\omega})\}_{i=1}^{W}$ provides a good indicator for the relative error ($\epsilon_{rel})$ of $M_0(e^{\mathbf{i}\omega})$ to $H_1(e^{\mathbf{i}\omega})$.
  • Figure 2: Frequency responses ((a) and (c)) and point-wise relative errors (plots (b) and (d)) of DDROMs approximating $H_2$, constructed using frequency data recovered from time-domain data via \ref{['alg:DataInform']} ((a) and (b)) and true frequency data ((c) and (d)).
  • Figure 3: Error in $M_0(\sigma)$ (stars) and the boundary of the set of points one standard deviation from $M_0(\sigma)-H_1(\sigma)$ (solid circles) for different values of $K$.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: burohmanBSC2020informativity
  • Remark 2.5
  • Lemma 3.1
  • Proof 1
  • Corollary 3.2
  • Corollary 3.3
  • Remark 3.4
  • ...and 4 more